April 30, 2010

Wise’s work on groups with quasi-convex hierachies

Two weeks ago, there was a very interesting conference in Montréal organized around Dani Wise’s work on subgroup separability properties for certain word-hyperbolic groups, namely those with “quasi-convex hierachies”. I’ve mentioned this work twicebefore. This time Dani gave about 12-15 hours of lectures. Stefan Friedl took notes and typed then up into a 21-page summary which you can read here.

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I think the conference turned out pretty well, but unfortunately about six people couldn’t come from
Europe due to the volcano, including two scheduled speakers. Nathan filled in for Bergeron and
gave a nice talk on the asymptotic number of fibered faces of the Thurston norm ball
of congruence covers of a certain arithmetic hyperbolic 3-manifold (joint with Ramakrishnan). Replacing Boileau,
I gave a survey talk of the state of the virtual Haken conjecture, and formulated some conjectures
related to Dani’s work.

I thought Dani’s talks were valuable, but reading his 200 page manuscript is somewhat daunting.
He shared a draft with the people who attended his talks, but he still needs to finish some parts of the
paper, and is waiting for some feedback from experts before distributing it more widely.
The full logic of the proof also incorporates the result of a 40 page preprint of Haglund and Wise, which is submitted,
and work of Hsu and Wise, which Hsu spoke about at the conference. In some sense, Dani’s
paper is a complex inductive argument to reduce the case of a general quasi-convex hierarchy
to the case of an almost malnormal quasi-convex hierarchy (using an extension of small-cancellation theory
to cube complexes, based on ideas of Casson). The almost malnormal quasi-convex hierarchy case is taken
care of by combining his joint works with Haglund and Hsu. In my view, the crux of the argument
(where in some sense residual finiteness is proven) is the Haglund and Wise paper. I haven’t yet
understood the argument in this paper, but I had some helpful discussions with Dani about it, and
the argument is closely modeled on the Dani’s paper on polygons of finite groups.